Identification

Title

Bayesian reduced-order deep learning surrogate model for dynamic systems described by partial differential equations

Abstract

We propose a reduced-order deep-learning surrogate model for dynamic systems described by time-dependent partial differential equations. This method employs space–time Karhunen–Loève expansions (KLEs) of the state variables and space-dependent KLEs of space-varying parameters to identify the reduced (latent) dimensions. Subsequently, a <a class="topic-link" style="background-color:rgba(0, 0, 0, 0);box-sizing:border-box;color:rgb(31, 31, 31);margin:0px;padding:0px;text-decoration:underline 1px rgb(31, 31, 31);text-underline-offset:1px;word-break:break-word;" href="https://www.sciencedirect.com/topics/chemical-engineering/deep-neural-network" title="Learn more about deep neural network from ScienceDirect's AI-generated Topic Pages">deep neural network</a> (DNN) is used to map the parameter latent space to the state variable latent space.</div><div class="u-margin-s-bottom" style="-webkit-text-stroke-width:0px;box-sizing:border-box;color:rgb(31, 31, 31);font-family:ElsevierGulliver, Georgia, "Times New Roman", Times, STIXGeneral, "Cambria Math", "Lucida Sans Unicode", "Microsoft Sans Serif", "Segoe UI Symbol", "Arial Unicode MS", serif, sans-serif;font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;margin-bottom:16px !important;margin-left:0px;margin-right:0px;margin-top:0px;orphans:2;padding:0px;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;" id="d1e1441">An approximate Bayesian method is developed for uncertainty quantification (UQ) in the proposed KL-DNN surrogate model. The KL-DNN method is tested for the linear advection–diffusion and nonlinear diffusion equations, and the Bayesian approach for UQ is compared with the deep ensembling (DE) approach, commonly used for quantifying uncertainty in DNN models. It was found that the approximate Bayesian method provides a more informative distribution of the PDE solutions in terms of the coverage of the reference PDE solutions (the percentage of nodes where the reference solution is within the confidence interval predicted by the UQ methods) and log predictive probability. The DE method is found to underestimate uncertainty and introduce bias.</div><div class="u-margin-s-bottom" style="-webkit-text-stroke-width:0px;box-sizing:border-box;color:rgb(31, 31, 31);font-family:ElsevierGulliver, Georgia, "Times New Roman", Times, STIXGeneral, "Cambria Math", "Lucida Sans Unicode", "Microsoft Sans Serif", "Segoe UI Symbol", "Arial Unicode MS", serif, sans-serif;font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;margin-bottom:16px !important;margin-left:0px;margin-right:0px;margin-top:0px;orphans:2;padding:0px;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;" id="d1e1443">For the nonlinear diffusion equation, we compare the KL-DNN method with the Fourier Neural Operator (FNO) method and find that KL-DNN is 10% more accurate and needs less training time than the FNO method.

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document

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https://n2t.net/ark:/85065/d77085qt

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eng

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geoscientificInformation

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publication

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2016-01-01T00:00:00Z

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publication

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2024-09-01T00:00:00Z

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<span style="font-family:Arial;font-size:10pt;font-style:normal;" data-sheets-root="1">Copyright author(s). This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.</span>

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OpenSky Support

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UCAR/NCAR - Library

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PO Box 3000

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80307-3000

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opensky@ucar.edu

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http://opensky.ucar.edu/

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